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what about differences in 223 and 5.56?
There is a lot of conflicting info out there....

Well, in my experience .223 works fine in a 5.56 chambered weapon. 5.56 is a little snug in a .223 chamber. It leads to cycling issues & cases getting stuck in the chamber. The best happy medium I have found to date is what Rock River calls a Wylde chamber. It's kind of a blended chamber that I have had no issues with firing or cycling either round. If you are looking for a dead on target gun, get a barrel chambered for either specific round & fire only that round through it.

my understanding is the main difference is in the thickness of brass of the cartridges is different - with 556 being the thicker brass. So if one were to load a 223 poweder charge in a 556 case, ther could be some issues with pressure. If anyone knows different or can add to, please feel free to do so.

Does anyone here have a high powered scope on the top of their rifle and a red dot sight mounted at a 45 degrees angle in addition to that? What are some cons about doing that?

No problems at all if you rotate the rifle 45 deg when sighting it in and it puts the holo or red dot into normal sight position.
Not so good if you just move your eye to the holo without turning it to the topside. Windage is set, but goes off when it is not centered above barrel in the normal sight position.
Hope that made sense.

Does anyone here have a high powered scope on the top of their rifle and a red dot sight mounted at a 45 degrees angle in addition to that? What are some cons about doing that?

No problems at all if you rotate the rifle 45 deg when sighting it in and it puts the holo or red dot into normal sight position.
Not so good if you just move your eye to the holo without turning it to the topside. Windage is set, but goes off when it is not centered above barrel in the normal sight position when firing.
Hope that made sense.
I run a 3 sight system, Scope, Holo at 45 deg and QD mount and Troy flip ups..
My trust is in my scope and troys at long range and holo only for close quarter.
The holo really isn't needed.

I purchased a Stripped lower receiver: it is a Shadow Ops Model SHDW-15B, and is a 300 AAC. I know little about AR's, what does the 300 AAC mean? I have a kit to complete the lower, but what kind of upper do I need to get?

.300aac is a caliber (7.62×35mm) Not the traditional caliber for an AR-15 .223 or 5.56

Most guy with a 300 marked lower are building a dedicated 300 gun.

Your lower can be built with ANY standard ar-15 parts kit.
Your AR can also have any ar-15 (not ar10) caliber compatible upper including:

5.56
.223
.300aac
7.62×39mm (different mag needed)
Etc Etc

When you finish your rifle it will still say .300aac on the side. Leading people to believe it has a 300aac upper on it. If you put a .223 upper on it, it will be a little confusing.. But won't compromise any functionality.